162 IEEE ELECTRON DEVICE LETTERS, VOL. 31, NO. 2, FEBRUARY 2010
Investigation of Switching-Time Variations
for Nanoscale MOSFETs Using the
Effective-Drive-Current Approach
Yu-Sheng Wu, Student Member, IEEE, Ming-Long Fan, Student Member, IEEE, and Pin Su, Member, IEEE
Abstract—This letter investigates the impacts of random dopant fluctuation (RDF) and line edge roughness (LER) on the switching-time (ST) variation for nanoscale MOSFETs using the effective-drive-current (Ieff) approach that decouples the ST variation into transition-charge (ΔQ) and Ieff variations. Al-though the RDF has been recognized as the main variation source to the threshold-voltage variation, this letter indicates that the relative importance of LER increases as the ST variation is considered.
Index Terms—Line edge roughness (LER), MOSFET, random dopant fluctuation (RDF), switching time (ST).
I. INTRODUCTION
W
ITH MOSFET scaling, the impact of random dopant fluctuation (RDF) and line edge roughness (LER) on the threshold-voltage (Vth) variation of nanoscale transistors isgrowing and being extensively examined [1]–[3]. For example, Asenov et al. [2] has shown that, for a given LER variation, as gate dimensions are reduced, the Vth fluctuations increase
and are comparable in magnitude to those caused by RDF. Roy et al. [3] has concluded that the Vth variation due to RDF
would dominate the behavior of the bulk MOSFETs if the LER can meet the prescription of the International Technology Roadmap for Semiconductors.
For logic circuits, however, the variation of signal switching time (ST) due to RDF and LER is particularly important. Whether there is any gap between Vthand ST variations merits
investigation. In this letter, we investigate the ST variation due to RDF and LER for bulk MOSFETs using the approach of effective drive current in CMOS inverters [4].
II. METHODOLOGY
We decouple the ST variation into transition-charge (ΔQ) and effective-drive-current (Ieff) variations. The ST can
be defined as ΔQ/Ieff [5], where ΔQ is the transition Manuscript received June 2, 2009; revised October 30, 2009. First published December 22, 2009; current version published January 27, 2010. This work was supported in part by the National Science Council of Taiwan under Contract NSC 98-2221-E-009-178 and in part by the Ministry of Education in Taiwan under the ATU Program. The review of this letter was arranged by Editor K. De Meyer.
The authors are with the Department of Electronics Engineering and Insti-tution of Electronics, National Chiao Tung University, Hsinchu 30013, Taiwan (e-mail: pinsu@falculty.nctu.edu.tw).
Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/LED.2009.2037247
charge between logic “ON” and “OFF” states. The ΔQ
for an NFET can be calculated by Qn(VGS= VDD, VDS=
0 V)− Qn(VGS= 0 V, VDS= VDD). The Ieff for an NFET
can be approximated as [IDS(VGS= VDD, VDS= 0.5VDD) +
IDS(VGS= 0.5VDD, VDS= VDD)]/2 [4]. Therefore, in
con-trast to the time-consuming mixed-mode transient simulation, only dc simulation for a single device is needed to derive ΔQ and Ieff. More importantly, the effective-drive-current
approach may provide physical insights in the assessment of ST variations.
The device parameters of bulk MOSFETs used in this letter are gate length (Lgate) = 25 nm, channel width (W) = 25 nm,
oxide thickness (tox) = 0.8 nm, source/drain junction depth
(xj) = 12.5 nm, channel doping (Nch) = 4.8× 1018 cm−3,
and supply voltage (VDD) = 0.8 V. To assess the RDF in bulk
MOSFETs, we have carried out the atomistic device simulation using the Monte Carlo approach to generate the dopants in the channel [1]. To avoid the charge trapping in the sharp Coulomb potential well and, hence, the mesh size dependences of the simulation results, we have employed the density gradient method in our atomistic simulation [3]. The boundary condition at the Si/SiO2interface for the density gradient method is that
the carrier density changes continuously across the interface, i.e., the continuity of the wavefunctions across the interface [6]. Fig. 1(a) shows one of the 150 samples in our atomistic simulation. To assess the LER, the line edge patterns were de-rived using the Fourier synthesis approach similar to the one in [2], and then, the Monte Carlo simulation was performed. The parameters used in the LER simulation are the rms amplitude Δ = 1 nm [7] and the correlation length Λ = 30 nm. Fig. 1(b) shows one of the 150 samples in our simulation. In this letter, we use the drift-diffusion equation as the transport model. The velocity saturation model is used to assess the on-current under the high drain field.
III. RESULTS ANDDISCUSSION
Fig. 2(a) compares the saturation threshold-voltage (Vth,sat)
distributions due to RDF and LER for bulk MOSFETs. Fig. 2(b) compares the ST distributions due to the RDF and LER. It can be seen that the standard deviation of Vth,sat (σVth,sat) due to
RDF is larger than that due to LER. Nevertheless, Fig. 2(b) shows that the standard deviation of ST (σST) due to LER is comparable with that due to RDF. In other words, the relative importance of LER for ST variation increases as compared with
WU et al.: INVESTIGATION OF SWITCHING-TIME VARIATIONS FOR NANOSCALE MOSFETs 163
Fig. 1. Simulated bulk MOSFETs in this letter. (a) One of the samples with RDF and (b) one of the samples with LER.
Fig. 2. (a) Vth,satdistribution of 150 samples due to RDF and LER. (b) ST distribution of 150 samples due to RDF and LER.
Fig. 3. Normalized standard deviations of ST, Ieff, and ΔQ due to RDF and LER.
that for Vth,satvariation. Since ST = ΔQ/Ieff, the normalized
standard deviation of ST (σST/μST) can be approximated as
|σST/μST| ≈ |σΔQ/μΔQ − σIeff/μIeff|, where μST, μΔQ,
and μIeffare the mean values of ST, ΔQ, and Ieff, respectively.
In this letter, we consider the standard deviations σST, σΔQ, and σIeff as signed numbers. Fig. 3 shows the |σST/μST|,
|σΔQ/μΔQ|, and |σIeff/μIeff| (normalized standard deviation
of ST, ΔQ, and Ieff, respectively) caused by RDF and LER.
It can be seen that the |σST/μST| due to RDF is roughly equal to the difference of |σIeff/μIeff| and |σΔQ/μΔQ| due
to RDF. However, the |σST/μST| due to LER is roughly equal to the sum of |σIeff/μIeff| and |σΔQ/μΔQ| due to
LER. The results in Fig. 3 can be explained as follows. The impact of RDF on MOSFETs stems from the variation of the effective channel doping (Nch,eff). For devices with smaller
Nch,eff values, the Vthis smaller, and hence, Ieff and ΔQ are
larger because they are roughly proportional to (VGS− Vth).
Thus, Ieff and ΔQ are positively correlated [Fig. 4(a)].
There-fore, |σST/μST| is roughly equal to the difference between
|σΔQ/μΔQ| and |σIeff/μIeff| because the quantities of σΔQ
and σIeff have the same sign. In other words, the impacts of
RDF on ΔQ and Ieff are mutually canceled, and|σST/μST| is
reduced.
The impact of LER on MOSFETs results from the variation of the effective channel length (Leff). For devices with shorter
Leff’s, the Vth is smaller because of the short-channel effect,
and hence, the Ieff is larger. As for ΔQ, devices with shorter
Leff’s possess smaller ΔQ’s because ΔQ is proportional to
the gate area (W × Leff). Thus, Ieff and ΔQ are negatively
correlated [Fig. 4(b)]. Therefore, |σST/μST| is roughly equal to the sum of |σΔQ/μΔQ| and |σIeff/μIeff| because the
quantities of σΔQ and σIeff have the opposite sign. In other
words, the |σST/μST| is larger than either |σΔQ/μΔQ| or
|σIeff/μIeff|.
It should be noted that, for ultrascaled devices in which ballistic transport becomes significant, the Ieff will be
under-estimated if the drift-diffusion model is used. However, the normalized standard deviation of Ieff (σIeff/μIeff) may not be
164 IEEE ELECTRON DEVICE LETTERS, VOL. 31, NO. 2, FEBRUARY 2010
Fig. 4. Correlations of Ieffand ΔQ distributions for MOSFETs with (a) RDF and (b) LER.
as sensitive to the transport model as the Ieff. Aside from the
electrostatic potential fluctuation, the discrete dopants will also cause the mobility fluctuation, which is not considered in this
letter. Therefore, the σIeff due to RDF may be larger than that
in Fig. 3 because of the mobility fluctuation caused by discrete dopants.
IV. CONCLUSION
We have investigated the impacts of RDF and LER on the ST variation for nanoscale MOSFETs using the effective-drive-current approach that decouples the ST variation into ΔQ and
Ieffvariations. Our results indicate that the ST variation caused
by LER may be larger than that caused by RDF. This is because
Ieff and ΔQ variations due to RDF are mutually canceled and
the ST variation caused by RDF is reduced, while Ieff and
ΔQ variations due to LER increase the ST variation caused by LER. Although the RDF has been recognized as the main variation source to Vth variation, the relative importance of
LER increases as the ST variation is considered. This letter may provide insights for device and circuit designs using advanced CMOS technology.
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